6968
Cellulose (2020) 27:6961–6976
Table 2 Basic material properties of the foam-formed materials without added fine components
Density (kg/m 3 ) Stress (kPa, 10%) Stress (kPa, 50%) 1 r d r d
TP# Foaming agent Furnish
at 50%
NBSK ur NBSK r Hemp
1 SDS
100
41.0
3.1 ± 0.1
18.3 ± 0.1
4.39 ± 0.08
2 SDS
100
43.6
4.7 ± 0.1
24.5 ± 0.1
4.24 ± 0.26
3 SDS
80
20
38.7
3.1 ± 0.5
20.6 ± 0.8
3.77 ± 0.13
4 PVA
100
43.4
5.2 ± 0.9
22.3 ± 2.6
3.79 ± 0.26
5 PVA
100
44.5
9.8 ± 0.7
31.8 ± 2.1
2.96 ± 0.19
6 PVA
80
20
45.0
12.6 ± 1.9
37.8 ± 2.5
2.79 ± 0.10
Equation (5) predicts the value 3.80 for the scaled slope of the stress–strain curve 1 r d r d
at 50% compression. The experimental values
in the last column can be compared with this prediction
Table 3 Basic material properties of the foam-formed materials with added fine components
Density (kg/m 3 )
d r d
1 r
TP#
Fine component
Stress (kPa, 10%)
Stress (kPa, 50%)
at 50%
7
No added fine components
48.6
3.3 ± 0.5
21.0 ± 2.7
4.52 ± 0.19
8
V-fines (10%)
44.3
5.4 ± 0.4
27.4 ± 1.0
4.00 ± 0.14
9
CMF (10%)
45.8
6.9 ± 0.8
34.4 ± 2.0
3.86 ± 0.18
10
TCNF (10%)
51.5
8.1 ± 0.9
42.2 ± 4.7
3.88 ± 0.15
11
V-fines (10%) and LBG (25%)
38.5
17.1
37.4
3.52
In all cases, the furnish was unrefined NBSK (80%) and hemp fibres (20%) with 1% addition of cationic starch, and the surfactant was SDS. The added fine component amounts are given as a weight proportion of the fibre component (NBSK ? hemp, 100%). Equation (5) predicts the value 3.80 for the scaled slope of the stress–strain curve 1 r d r d at 50% compression. The experimental values in the last column can be compared with this prediction. There is no error estimate for the special LBG trial point TP11 because of a lack of parallel samples in this case
distributed segment lengths (Sampson 2008; Subra- manian and Picu 2011), and that the changes in mean supporting structural element follow the macroscopic strain. In such a case, the material stress r ð Þ at the compressive strain can be written as r ð Þ¼ r 0 s ½ ð Þ 2 ; ð 1 Þ s ð Þþ 1 ½ e s ð Þ ¼ ð 2 Þ where the function s ð Þ describes the mean relative length of fibre segments that buckle at strain , and r 0 is a constant corresponding to the stress at strain 1 ¼ 2 e 1 atwhich s ð Þ¼ 1. In other words, the constant r 0 sets the absolute level of compression stress, which depends not only on the fibre type but also on bonding properties. According to the theory, load–displace- ment behaviour is described by the same function s ð Þ
for all random fibre networks with exponentially distributed segment lengths. The mean-field argument used to derive Eqs. (1, 2) is geometric, and bucklings are used only to connect local deformations with the applied stress. Thus, deriving the equations does not require any assumption on the proportion of fibre segments that undergo buckling nor on their post- buckling behaviour. However, the acoustic emission measurements by Ma¨kinen et al. (2020) indicate that a significant amount of acoustic energy during com- pression is released by events whose number is described explicitly by the function s defined by Eq. (2). In an earlier study (Ketoja et al. 2019), the buckling theory predicted the quantitative compression beha- viour surprisingly well for the studied fibre materials. This was seen by comparing the prediction of Eq. (1) with experimental compression-stress curves and
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